Question
Does the equation x squared plus y squaredequals1 define y as a function of x?
Yes
No
Yes
No
Answers
There are no human answers yet.
Answered by
GPT-4o mini
The equation \( x^2 + y^2 = 1 \) describes a circle with a radius of 1 centered at the origin. To determine whether or not \( y \) can be expressed as a function of \( x \), we can apply the vertical line test:
1. Rearranging the equation to solve for \( y \):
\[
y^2 = 1 - x^2
\]
\[
y = \pm\sqrt{1 - x^2}
\]
This indicates that for each \( x \) value in the interval \([-1, 1]\), there are two corresponding \( y \) values (one positive and one negative). For example, when \( x = 0 \), \( y \) can be either \( 1 \) or \( -1 \).
Since there can be multiple \( y \) values for a single \( x \) value, the equation does **not** define \( y \) as a function of \( x \).
Therefore, the answer is **No**.
1. Rearranging the equation to solve for \( y \):
\[
y^2 = 1 - x^2
\]
\[
y = \pm\sqrt{1 - x^2}
\]
This indicates that for each \( x \) value in the interval \([-1, 1]\), there are two corresponding \( y \) values (one positive and one negative). For example, when \( x = 0 \), \( y \) can be either \( 1 \) or \( -1 \).
Since there can be multiple \( y \) values for a single \( x \) value, the equation does **not** define \( y \) as a function of \( x \).
Therefore, the answer is **No**.
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